Finite Group Databasebeta

← #192452

Subgroups — #192452

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21986C211yesmaximal, characteristic
3#21866C211yesmaximal, characteristic
4C2×#1515C211yesmaximal, characteristic
5#14298C211yesmaximal, characteristic
6#8872C214yesmaximal
7#8868C214yesmaximal
8#8018C211yesmaximal, characteristic
9#7912C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329C2214yes—
20#2317—21no—
21C4×C22⋊Q8—22no—
22#2285C2211yescharacteristic
23C2×#383C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#2274—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1993—21no—
28#1960—21no—
29#1949—22no—
30#1872C2212yes—
31#1872C2212yes—
32#1853C2212yes—
33#1852C2212yes—
34#1847—21no—
35#1836—21no—
36#1805—22no—
37#1515C418yescomplemented
38C2×#328C2214yes—
39#1444—21no—
40#1444C2212yes—
41#1442—21no—
42#1442C2212yes—
43#1436—22no—
44#1294C2214yes—
45#1280C2214yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#432—22no—
70#432C2×C414yes—
71C2×#107—22no—
72C2×#106—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×#103—22no—
76C2×#103—22no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×#97—21no—
80C2×#97—21no—
81#391—22no—
82#391—24no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#385—24no—
91#384—22no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#383—22no—
95#383C2×C414yes—
96#382—22no—
97#382—21no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382—21no—
102#380—21no—
103#380C2312yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4—24no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95C2312yes—
125C2×#95C2312yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#328C2×C4116yes—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8C4⋊C414yescomplemented
146C4⋊Q8C4⋊C414yescomplemented
147#125—24no—
148#125—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8C22×C414yes—
154C22⋊Q8C2×Q814yes—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—84no—
159C4×Q8C4⋊C418yescomplemented
160#117—24no—
161#117—24no—
162#117—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—82no—
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209#107—82no—
210#106—82no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213#103—82no—
214#103—82no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8C22×C414yes—
219#97—24no—
220#97—42no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—44no—
237#95—82no—
238#95—82no—
239#95C22×C414yes—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8C2×C4⋊C414yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4#13714yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q811yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—88no—
328C4⋊C4C2×C4⋊C414yes—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—44no—
332C42—28no—
333C42—44no—
334C42—44no—
335C42—82no—
336C42—84no—
337C42—84no—
338C42C2×C4⋊C412yes—
339C42C2×C4⋊C412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42C22⋊Q812yes—
343C42C22⋊Q812yes—
344C42#12312yes—
345C42#12312yes—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23C2×C4×D411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×C22⋊Q811yescharacteristic
353C23C2×#13712yes—
354C23#57512yes—
355Q8—44no—
356Q8—48no—
357Q8—82no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4#38512yes—
417C2×C4#38512yes—
418C2×C4#38512yes—
419C2×C4#38512yes—
420C2×C4C22×C4⋊C411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×#12311yescharacteristic
424C8—44no—
425C8—82no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22C2×#38511yescharacteristic
435C22C4×#13712yes—
436C22#300411yescharacteristic
437C22#300511yescharacteristic
438C22#301911yescharacteristic
439C22#302011yescharacteristic
440C22#302711yescharacteristic
441C22#302811yescharacteristic
442C22#307812yes—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—84no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4—161no—
459C4—161no—
460C2—22no—
461C2—22no—
462C2#1836411yescharacteristic
463C2#1836511yescharacteristic
464C2#1836911yescharacteristic
465C2#1837011yescharacteristic
466C2#2086411yescharacteristic
467C2#2087411yescharacteristic
468C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
37. #1515C4semidirect
145. C4⋊Q8C4⋊C4semidirect
146. C4⋊Q8C4⋊C4semidirect
159. C4×Q8C4⋊C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#342132
#219862
#218662
C2×#15152
#142982
#88722
#88682
#80182
#79122